Enumeration of interval graphs and $d$-representable complexes

TL;DR

This paper provides asymptotic estimates for counting $d$-representable simplicial complexes on $n$ vertices, including new results for interval graphs, and shows they form a negligible fraction of $d$-collapsible complexes.

Contribution

It offers the first asymptotic enumeration of $d$-representable complexes for fixed $d$, extending known results for interval graphs.

Findings

Asymptotic formulas for $d$-representable complexes

New enumeration results for interval graphs ($d=1$)

Demonstrates $d$-representable complexes are rare among $d$-collapsible complexes

Abstract

For each fixed $d\ge 1$, we obtain asymptotic estimates for the number of $d$-representable simplicial complexes on $n$ vertices as a function of $n$. The case $d=1$ corresponds to counting interval graphs, and we obtain new results in this well-studied case as well. Our results imply that the $d$-representable complexes comprise a vanishingly small fraction of $d$-collapsible complexes.